Intersection cohomology of the circle actions

Abstract

A classical result says that a free action of the circle S 1 on a topological space X is geometrically classified by the orbit space B and by a cohomological class e ∈ H 2 (B, Z), the Euler class. When the action is not free we have a difficult open question: Π : "Is the space X determined by the orbit space B and the Euler class?"The main result of this work is a step towards the understanding of the above question in the category of unfolded pseudomanifolds. We prove that the orbit space B and the Euler class determine:• the intersection cohomology of X,• the real homotopy type of X.

Equivariant intersection cohomology of the circle actions

“…The localization of the equivariant intersection cohomology is a cohomological theory introduced in [1,6]. In fact, it is a residual cohomology since it depends on a neighborhood of the fixed point set F. The usual 10 LocalizationTheorem establishes that the localization L * The Λe-product is induced by e·…”

Categorical Properties Of Smooth Unfoldings On Stratified Spaces

“…(4) Since algebraic manifolds satisfy the Withney's conditions, every algebraic manifold is a stratified pseudomanifold [15]. (5) The orbit space of a stratified pseudomanifold endowed with a suitable stratified action of a compact Lie group is again a stratified pseudomanifold [14,16]. (6) The foliation space of a suitably controlled locally conic foliated manifold is a stratified pseudomanifold [17,19].…”

“…✲ B is unfoldable morphism [10]. A completely analogous situation can be given for any stratified pseudomanifold that supports a suitable stratified action [14].…”

Intersection homology and Poincaré duality on homotopically stratified spaces

“…To name just a few: Right in [21], Goresky and MacPherson introduced signatures and L-classes for pseudomanifolds with only even codimension strata; Siegel extended signatures and bordism theory to Witt spaces [42]; and various extensions of duality and characteristic classes have been studied by Banagl, Cappell and Shaneson in various combinations [1; 2; 4; 10; 13]. The application of intersection homology to the study of group actions both on smooth manifolds and on stratified spaces is also an active field of research; see, eg, Brylinski [9], Cappell, Shaneson and Weinberger [13], 1 excluding those with codimension one strata Curran [15], Hector and Saralegi [25], Padilla [36], Padilla and Saralegi-Aranguren [37] and Saralegi-Aranguren [40]. For more on applications of intersection homology in these directions, as well as in other fields, we refer the reader to the expository sources Banagl [3], Kirwan [34] and Kleiman [35].…”

“…A surgery theory for MHSSs has been developed by Weinberger [38], and their geometric neighborhood properties have been studied by Hughes, culminating in [23]. In [33], Quinn noted that MHSSs "are defined by local homotopy properties, which seem more appropriate for the study of a homology theory" than the local homeomorphism properties of pseudomanifolds, and he showed that intersection homology is a topological invariant on 1 excluding those with codimension one strata 2 The application of intersection homology to the study of group actions both on smooth manifolds and on stratified spaces is an active field of research; see, e.g., [20,35,30,10,32,31].…”